Union Distributes over Intersection/Family of Sets/Corollary
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Theorem
Let $I$ and $J$ be indexing sets.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\beta}_{\beta \mathop \in J}$ be indexed families of subsets of a set $S$.
Then:
- $\ds \bigcap_{\tuple{\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cup B_\beta} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcap_{\beta \mathop \in J} B_\beta}$
where $\ds \bigcap_{\alpha \mathop \in I} A_\alpha$ denotes the intersection of $\family {A_\alpha}_{\alpha \mathop \in I}$.
Proof
\(\ds \bigcap_{\alpha \mathop \in I} \paren {A_\alpha \cup B}\) | \(=\) | \(\ds \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B\) | Union Distributes over Intersection of Family | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \bigcap_{\alpha \mathop \in I} \paren {A_\alpha \cup \paren {\bigcap_{\beta \mathop \in J} B_\beta} }\) | \(=\) | \(\ds \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcap_{\beta \mathop \in J} B_\beta}\) | setting $\ds B = \paren {\bigcap_{\beta \mathop \in J} B_\beta}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \bigcap_{\alpha \mathop \in I} \paren {\bigcap_{\beta \mathop \in J} \paren {A_\alpha \cup B_\beta} }\) | \(=\) | \(\ds \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcap_{\beta \mathop \in J} B_\beta}\) | Union Distributes over Intersection of Family | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \bigcap_{\paren {\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cup B_\beta}\) | \(=\) | \(\ds \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcap_{\beta \mathop \in J} B_\beta}\) |
$\blacksquare$
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 9$: Families